COMBINING PARTIAL EVALUATION AND SYMBOLIC EXECUTION
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1 COMBINING PARTIAL EVALUATION AND SYMBOLIC EXECUTION Reiner Hähnle & Richard Bubel Chalmers University Symposium 09 Speyer
2 CONTROL CIRCUIT y = 80; threshold = 100; if (y > threshold) { decrease = true; } else { decrease = false; } while ( y-threshold > eps) { y = y-1 : y+1; }
3 CONTROL-FLOW GRAPH y = 80 threshold=100 y>threshold? decrease = true; decrease = false; y-threshold > eps? y = 80; threshold = 100; if (y > threshold) { decrease = true; } else { decrease = false; } while ( y-threshold > eps) { y = y-1 : y+1; }
4 PARTIAL EVALUATION decrease = true; y = 80 threshold=100 y>threshold 80>100 false?? decrease = false; Evaluator y 80 threshold 100 decrease false Static information propagated along CFG constant propagation constant expression evaluation y-threshold y-100 > eps > eps?? decrease false? Evaluator y - threshold 100 decrease false dead-code elimination other: type coercion, safe dereferencing etc.
5 PARTIAL EVALUATION A Bit More Realistic y = 80 threshold=100 y>threshold? threshold=100 y>100? decrease = true; decrease = false; decrease = true; decrease = false; y-threshold > eps? y-100 > eps?
6 SYMBOLIC EXECUTION threshold=100 y>threshold? decrease = true; decrease = false; y-threshold > eps?
7 SYMBOLIC EXECUTION threshold=100 y>threshold? threshold=100 decrease = true; decrease = false; y>threshold? y-threshold > eps? decrease = true; decrease=false; y-threshold > eps? y-threshold > eps? y-threshold > eps? y-threshold > eps? y-threshold > eps? y-threshold > eps?
8 OPTIMIZING SYMBOLIC EXECUTION Symbolic Execution unfolds control-flow graph into tree unfeasible paths must be closed by first-order proof search interleave partial evaluation and symbolic execution
9 INTERLEAVING threshold=100 y>threshold? decrease = true; decrease = false; threshold=100 y>threshold y>100?? y-threshold > eps? decrease=true; decrease=false; y-threshold y-100 > eps > eps?? y-threshold y-100 > eps > eps?? true decrease false? y-threshold y-100 > eps > eps?? decrease true? y-threshold y-100 > eps > eps?? y-threshold y-100 > eps > eps?? y-threshold y-100 > eps > eps?? decrease true? decrease false? decrease false?
10 INTERLEAVING threshold=100 threshold=100 y>threshold? decrease = true; decrease = false; decrease=true; y>100? decrease=false; y-threshold > eps? y-100 > eps? y-100 > eps? true false y-100 > eps? y-100 > eps? true false
11 PROGRAM LOGIC Programming Language Simple OO-Programming Language single inheritance dereferencing null, division by zero etc. cause nontermination dynamic method binding no nested expressions
12 PROGRAM LOGIC Syntax Dynamic Logic with Updates: (as usual) Specialisation operator : P rgel Upd F or P rgel where P rgel = Statement Expression p (U, ϕ) denotes a program equivalent to if is executed in a state s satisfying ϕ p p and coinciding on U Examples: x =(y) (y := 3, true)+3; (x = o.a + 3) (o.a := 10,o!= null)
13 PROGRAM LOGIC Notions Signature Σ: Program variables and attributes are modelled as non-rigid constants and unary function symbols First-order structure(d, I): Domain D: sorted universe (interpretes sorts) Interpretation I: interpretes rigid function and predicate symbols States s S: interpretes program variables and attributes
14 PROGRAM LOGIC Signature Extension Partial Evaluation may extend the signature (temporary variables, anonymous updates ) p Σ Σ (U, ϕ) where (D, I) (D, I) and s s and β β Σ Σ Σ Σ Σ Σ
15 PROGRAM LOGIC Soundness Condition on the Specialisation Operator p Σ Σ (U, ϕ) For all formulas ψ over Σ, for all (D, I),s, β : Σ Σ Σ (D, I),s, β = Σ Σ Σ ( ) p (U, ϕ) ψ {U}(ϕ p ψ)
16 PROGRAM LOGIC Partial Evaluation Rules Rewrite Action Correctness Requirement Dead-Code Elimination if(b){p}else{q} (U, ϕ) p (U, ϕ) U(ϕ b. = true) Safe Field Access Partial Evaluator Propagation o.a (U, (U, ϕ) (p; q) (U, ϕ) p (U, ϕ); q (U, ϕ ) U(ϕ!(o. = null)) respmodstrong(p, mod) U := UV mod (D, I) = {U}{V mod }ϕ (D, I) = {U} p ϕ
17 PROGRAM LOGIC Partial Evaluator Introduction Rules Γ U!(o. = null), Γ U{o.a := t} q (o.a := t,!o. = null) φ, Γ U o.a = t; q φ, and several others
18 PROGRAM LOGIC Type Inference Rules res = o.m(a 1,..., a n ) (ϕ, U) o = o (ϕ, U) U(ϕ o!. = null & C :: instance(o)) res ).m(a 1 (ϕ, U),..., a n (ϕ, U))
19 PROGRAM LOGIC Type Inference Rules A a =...; boolean eq = a.equals(c); A a is an instance of C B C Evaluator...
20 DEMO
21 FUTURE WORK Simplification of specifications Partial evaluation of contracts and loop invariants Applicable to JavaCardDL / JML / OCL Investigate applicability to application engineering
22 FUTURE WORK Application Engineering Model Driven Architecture Platform Independent Model Platform Definition Model Partial Evaluation Program p (U, ϕ) Application Engineering Productline Artefacts Feature Configuration Platform Specific Model p (U, ϕ) Application
23 CONCLUSION (for the moment) y = 80; threshold = 100; if (y > threshold) { decrease = true; } else { decrease = false; } while (y-threshold > eps) { β y = y-1 : y+1; } partial eval. as replaced computation generalisation of proof search linear in β-reduction by computation number of locs in Hoare/VCG
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